 
Summary: arXiv:math.DG/0701028v131Dec2006
EXTREMAL METRICS ON BLOW UPS
C. AREZZO, F. PACARD, AND M. SINGER
1. Introduction
In this paper we study the problem of constructing extremal Kšahler metrics on blow ups at
finitely many points of Kšahler manifolds which already carry an extremal metric.
In [8], [9] Calabi has proposed, as best representatives of a given Kšahler class [] of a complex
compact manifold (M, J), a special type of metrics baptized extremal. These metrics are criti
cal points of the L2
square norm of the scalar curvature s. The corresponding EulerLagrange
equation reduces to the fact that
s := J s + i s
is a holomorphic vector field on M. In particular, the set of extremal metrics contains the set
of constant scalar curvature Kšahler ones. Calabi's intuition of looking at extremal metrics as
canonical representatives of a given Kšahler class has found a number of important confirmations
and also (unfortunately) nontrivial constraints. Calabi himself proved that an extremal Kšahler
metric must have the maximal possible symmetry allowed by the complex manifold M, and,
as observed by LeBrun and Simanca [17], this symmetry group can be fixed in advance. More
precisely, the identity component of the isometry group of any extremal metric g must be a
maximal compact subgroup of Aut0(M, J), the identity component of the group Aut(M, J) of
